<<Can malware remain on the boot sector after a full reinstall?>>
No...a clean install generates a brand new boot sector/MBR.
<<Is there any other place that malware could remain? (I only have the one partition).>>
Not on that hard drive. Of course, if you have USB/flash drives attached...they can also be infected.
<<And finally, what would I do to rewrite the boot sector to be sure that there is nothing bad left there.>>
Perform a clean install, following the suggestions outlined in the following links:
Clean Install Procedure with Illustrative Screen Captures - http://www.theeldergeek.com/xp_home_install_-_graphic.htm
XP Clean Install, Stevens - http://www.michaelstevenstech.com/cleanxpinstall.html
FWIW: Bear in mind that I also think it's wise to delete any existing partitions that have already been created...and start by creating a new partition (using the XP install CD). If that is done, any problems with the partition or existing file structure...which contributed to system/owner unhappiness...will be eliminated.
Of course, nothing will overcome hardware (memory, hard drive, etc.) problems which are significant, other than a change of hardware.